[EM] Wasted votes in party list, and cloneproofing
Kristofer Munsterhjelm
km_elmet at t-online.de
Fri Aug 19 15:44:35 PDT 2016
Suppose we have a party list method, and for simplicity's sake suppose
it's quota based rather than divisor based. Also suppose part of the
result goes like this:
Party L1: 3.3 seats
Party L2: 2.3 seats
Party L3: 1.3 seats
Party L4: 0.1 seats
and that the largest remainders are distributed so that all these L
parties get their seat shares rounded down. It then seems like there are
some wasted votes. Suppose the voters compromised instead, voting:
Party L1: 4 seats
Party L2: 2 seats
Party L3: 1 seats
Party L4: 0 seats
Now the L bloc has got one more seat by their voters compromising so
that the wasted shares end up giving L1 another seat.
I'd like to have a criterion that somewhat formalizes whether a method
is vulnerable to that kind of compromising incentive, since the
scenarios could potentially get very complex. Suppose the L-voters
compromise like this, then some other bloc's voters (say R-voters)
compromise in return; would that render the L-voters' strategy
But then, after thinking for a bit, I thought that the independence of
clones criterion would work. Suppose that the L-bloc all vote the
L-parties ahead of everybody else, i.e.
x_0: L1>L2>L3>L4>[others]
x_2: L1>L2>L4>L3>[others]
...
x_24: L4>L3>L2>L1>[others]
and that similarly, all voters who rank one of the L parties at position
p, rank all of them before ranking someone else,
then we could define a party list independence of clones criterion:
"Cloning a party should not affect the number of seats given to the
other parties, assuming that every party fields at least as many
candidates as there are seats". By pigeonhole, if any of the clones lose
or gain seats compared to the pre-clone party, those seats will be
acquired or lost by some party not part of the cloning process. The
second clause prevents false positives from the parties having too few
candidates to claim all their seats.
And it seems that if a method is cloneproof, the problem above can't
happen, as long as every voter rank the parties of the bloc next to each
other.
Suppose we have
Party L1: 3.3 seats -> 3 seats
Party L2: 2.3 seats -> 2 seats
Party L3: 1.3 seats -> 1 seat
Party L4: 0.1 seats -> 0 seats
Now, declone the L bloc to a single L party. Either L will have 6 seats
or it won't. If it doesn't, then we have a clone failure when we run the
scenario in reverse: cloning L into {L1, L2, L3, L4} made the number of
seats change. Call that failure "failure type 1".
So suppose it didn't change. Then we can clone L into {L1, L2, L3, L4}
so that
Party L1: 4 seats
Party L2: 2 seats
Party L3: 1 seats
Party L4: 0 seats
happens, which means that cloning paid off in that L got an additional
seat. Call that "failure type 2".
So if it's possible to redistribute the votes to get more seats, the
method will fail the independence of clones criterion - *if every voter
ranks the L-parties adjacent to each other*. Either the method will
exhibit failure type 1, type 2, or both.
This doesn't make the method immune to compromise just like the
independence from clones criterion doesn't make single-winner methods
immune to compromise; but at least a significant part of a "wasted bloc"
effect will be detectable by independence of clones failure.
Clearly, Plurality style party list (Webster, etc) fails the
independence of clones criterion. Let there be two parties (A and B)
with a 50-50 split. Then clone B into as many parties as there are
voters, and each voter votes for one of them. Unless there are very many
seats, A will get all of them. Or just consider that the one-seat
instance becomes Plurality, and Plurality is well known for clone problems.
Similarly, any procedure where you run a method to get scores for each
party (and this method doesn't take the number of seats as input), and
you then run a divisor method on the scores to apportion seats, must
fail. This because the base method doesn't know how many seats there are
and so can't quantize properly. In essence, virtual voters cast
Plurality-style ballots based on the first method so that the sum of
votes for each party is that party's score. Since manipulating the
Plurality-style ballots can be beneficial, so can manipulating the score
be, and the second method can't counter this because it doesn't know
about anything other than the scores provided by the first method (and
the number of seats).
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